Typically students learn what is known in science while scientists study the unknown. An area that is starting to be explored is what is unknowable in principle.
Staring with the seminal papers of Godel and Turing, this century has witnessed a stream of impossibility results, including undecidability, and intractability. But these results concern formal systems. Do they limit scientific knowledge?
I first spoke about these issues at a panel discussion in memory of the physicist Heinz Pagels on February 1, 1989, and at the Second Annual Charles Babbage Foundation Lecture in October, 1989. My talk at the 25th Anniversary of the Computer Science Department, Carnegie-Mellon University, delivered in 1990, appears in the Proceedings. See What is Scientifically Knowable?. Proceedings, Twenty Fifth Anniversary Symposium, School of Computer Science, Carnegie-Mellon University, Addison-Wesley, 1991, 489-503. One scientific problem that Iíve considered is protein folding. This is something that nature does easily but which we are unable to simulate and which theory suggests is very difficult to do. Possible reasons for this dissonance are presented in On Reality and Models, in Boundaries and Barriers: On the Limits to Scientific Knowledge, (J. Casti and A. Karlqvist, eds.), Addison-Wesley, 1996, 238-251.
Other papers include:
In May, 1996, there was a workshop at the Santa Fe Institute on Fundamental Sources of Unpredictability. Ten of the papers appeared as a special Proceedings issue of Complexity, Vol. 3, No. 1, 1997. The Proceedings Editors were J. Hartle, P. Hut, and J. F. Traub.
There are two formal systems that concern us. One is the mathematical model which is chosen by the scientist. Continuous models are common in fields varying from physics to economics. The real or complex number field is assumed. The second formal model is the model of computation. Computer scientists tend to favor the Turing Machine model. For scientific problems the real-number model has a number of advantages. The pros and cons of the real-number model versus the Turing Machine model are given in Chapter 8 of Complexity and Information.